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1
2 DIFFERENTIAL GEOMETRY
3 PROCEEDINGS OF THE THIRD SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held at the University of Arizona Tucson, Arizona February 18-19, 1960 With the Support of the NATIONAL SCIENCE FOUNDATION CARL B. ALLENDOERFER EDITOR
4 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME III D I F F E R E N T I A L G E O M E T R Y AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1961
5 Library of Congress Catalog Card Number Prepared by the American Mathematical Society under Grant No. NSF-G10809 with the National Science Foundation Copyright 1961 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers.
6 CONTENTS INTRODUCTION PAGE A Report on the Unitary Group 1 By RAOUL BOTT Vector Bundles and Homogeneous Spaces 7 By M. F. ATIYAH and F. HIRZEBRUCH A Procedure for Killing Homotopy Groups of Differentiable Manifolds. 39 By JOHN MILNOR Some Remarks on Homological Analysis and Structures By D. C. SPENCER Vector Form Methods and Deformations of Complex Structures By ALBERT NIJENHUIS Almost-Product Structures 94 By A. G. WALKER Homology of Principal Bundles 101 By ELD ON DYER and R. K. LASH OF Alexander-Pontrjagin Duality in Function Spaces 109 By JAMES EELLS, JR. The Cohomology of Lie Rings 130 By RICHARD S. PALAIS On the Theory of Solvmanifolds and Generalization with Applications to Differential Geometry 138 By Louis AUSLANDER Homogeneous Complex Contact Manifolds 144 By WILLIAM M. BOOTHBY On Compact, Riemannian Manifolds with Constant Curvature. I By EUGENIO CALABI Elementary Remarks on Surfaces with Curvature of Fixed Sign By L. NlRENBERG Canonical Forms on Frame Bundles of Higher Order Contact 186 By SHOSHICHI KOBAYASHI On Immersion of Manifolds 194 By HANS SAMELSON Index. 197 vii
7 INTRODUCTION This Symposium on Differential Geometry was organized as a focal point for the discussion of new trends in research. As can be seen from a quick glance at the papers in this volume, modern differential geometry to a large degree has become differential topology, and the methods employed are a far cry from the tensor analysis of the differential geometry of the loso's. This development, however, has not been as abrupt as might be imagined from a reading of these papers. It has its roots in the movement toward differential geometry in the large to which mathematicians such as Hopf and Rinow, Cohn-Vossen, de Rham, Hodge, and Myers gave impetus. The objectives of their work were to derive relationships between the topology of a manifold and its local differential geometry. Other sources of inspiration were E. Cartan (whose fundamental contributions were recognized by many only after his death) and M. Morse and his calculus of variations in the large. One of the major new ideas was that of a fiber bundle which gave a global structure to a differentiable manifold more general than that included in the older theories. Methods and results of differential geometry were applied with outstanding success to the theories of complex manifolds and algebraic varieties and these in turn have stimulated differential geometry. The discovery by Milnor of invariants of the differential structure of a manifold which are not topological invariants established differential topology as a discipline of major importance. GAEL B. ALLENDOERFER University of Washington, Seattle, Washington Vll
8 Ct-adic topology, 24 Affine connections, 94,186 Affine spaces, locally, 142 Alexander-Pontrjagin duality, 109 theorem, 124 Almost complex, 22 Almost-product structure, 94 Artin-Rees lemma, 24 Atlas Eulerian, 159 Lagrangian, 160 Axioms of a cohomology theory, 14 Bidifferentiable transformations, closed pseudogroups of, 61 Borsuk's Extension Theorem, 120 Bott isomorphism, 13 periodicity, 7 Bundles complex vector, 8 homology of principal, 101 fc-trivial, 49 of r-frames, 188 orien table, 115 ring of complex vector, 7 transverse, 115 C-space, 146 Ci-map, 20 Canonical form differential, 189 structure equation of, 191 Cartan, E. invariant forms' for a continuous pseudogroup of differentiate transformations, 85 structure equations of, 186 Category, model, 59 Characteristic class, 102 relative, 102 Chern character, 15, 29 x-equivalent, 40, 46, 49, 53, 55 Classification theorem, 29 Classifying spaces, 7, 28 Clifford-Klein spaces, 156 differentiable family of, 159 Closed pseudogroups of bidiffenertiable transformations, 61 INDEX 197 Cobordism, 40 Cochains, invariant, 135 Cohomology, 109 group of a Lie d-ring, 137 of Lie rings, 130 operations, 18 with values in the sheaves of Lie algebras of infinitesimal groups, 56 with values in sheaves of nonabelian groups, 56 Cohomology theory axioms of, 14 periodic, 7 Compact, Riemannian manifolds, 155 Complete germ, 72 Completed representation ring of a torus, 26 Completions of modules, 24 Complex, almost, 22 Complex analytic differentiable r-manifold, 64 family of complex structures, 92 Complex contact manifold, 144 manifold, homogeneous, 146 structure, 144 Complex structures complex analytic family of, 92 deformation of, 87 equivalence of, 89 family of, 91 obstructions to deformation of, stability of, 90 variations of, 89 Complex vector bundles, 8 ring of, 7 Connections affine, 94 linear (affine), 186 Constant curvature, Contact form, 145 Continuous pseudogroup of differentiable transformations, 81 invariant Cartan forms for, 85 Coordinate transformation, 118 Co-orienting, 111 Curvature, 186 constant, Gauss, 181
9 198 INDEX d-trivial Lie d-ring, 131 Deformations, homological analysis of, 69 Deformations of complex structures, 87 obstructions to, Deformation of the T-manifold, 70 germ of, 71 Derivative lie, 134 torsional, 99 Differentiability Graves-Hildebrandt, 115 Differentiable complex analytic or real analytic T-manifold, 64 family of T-manifolds, 70 Differentiable transformations continuous pseudogroups of, 81 invariant Cartan forms for a continuous pseudogroup of, 85 Differential form, 189 manifolds, 39 Distributions, 94 Double exterior forms, 166 Duality theorem, 110 Alexander-Pontrjagin, 124 Eilenberg and Steenrod axioms, 7 Equivalence of complex structures, 89 Eulerian atlas, 159 Existence in homological analysis, problem of, 75 Extension Theorem of Borsuk, 120 Exterior forms, double, 166 /-relatedness for vector forms, 90 Family of complex structures, 91 complex analytic, 92 Family of r-manifolds, differentiable, 70 Frames, r-, 188 bundle of, 188 Function spaces, 109 Fundamental class of the oriented pair, 113 r-manifolds deformation of, 70 differentiable family of, 70 differentiable, real analytic or complex analytic, 64 germ of deformation of, 71 T-structure, 64 T-vectorfield,67 Gauss curvature, 181 Germ, complete, 72 effective, 72 of deformation of the r-manifold, 71 stable, 75 Gradient mapping, 182 Grating, 112 Graves-Hildebrandt differentiability, 115 Groups killing homotopy, 39, 50 sheaf of, 65 unitarv, 1, 8 Weyl,23 Gysin homomorphism, 20, 114 Hildebrandt-Graves differentiability, 115 Homogeneous complex contact manifold, 146 spaces, 7, 31 Homological analysis of deformations, 69 Homology of principal bundles, 101 Homomorphism, Gysin, 20, 114 Homotopy complements, 113 killing classes, 43 killing groups, 39, 50 Hypersurfaces, 181 Immersion of manifolds, 194 Implicit function theorem, Infinitesimal pseudogroup, 66 Infinitesimally surjective, 75 Interior product, 133 Invariant Cartan forms for a continuous pseudogroup of differentiable transformations, 85 cochains, 135 cohomology group of a Lie coring, 137 Invariants, ring of, 27 Isomorphism Bott, 13 Theorem of Leray, 111 r. (/) source of, 187 target of, 187 Jacobi identities for vector forms, 88 Jet, r-, 187 A;-parallelizable manifold, 49 fc-trivial bundle, 49
10 INDEX 199 Killing homotopy classes, 43 groups, 39, 50 Klien-Clifford spaces, 156 differentiable family of, 159 < -ring, 132 Lagrangian atlas, 160 Leray Isomorphism Theorem, 111 Lie d-ring cohomology group of, 137 d-trivial, 131 invariant cohomology group of, 137 cc-module over, 132 over R, 131 Lie derivatives, 134 Lie group, compact, 25 connected, 23, 29, 36 representation ring of, 25 Lie rings, cohomology of, 130 Linear (affine) connection, 186 Locally afiine spaces, 142 stable, 74 trivial, 74 Manifold pair, orientation sheet of, 111 Manifolds, 39 compact, Riemannian, 155 complex contact, 144 deformation of the T-, 70 differentiable family of T-, 70 differentiable, real analytic or complex analytic T-, 64 germ of deformation of the T-, 71 homogeneous complex contact, 146 immersion of, 194 fc-parallelizable, 49 Mapping gradient, 182 monotone, 182 spherical image, 181 Model category, 59 Module over a Lie d-ring <, 132 basic, 132 cohomology of < with coefficients in, 136 invariant cohomology of with coefficients in, 136 impairing of two, 132 Modules completions of, 24 Monotone mappings, 182 Morse theory, 2 Multifoliate, 81 Multiplication, Pontrjagin, 125 Nilmanifold, 138 Noetherian ring, 24 Normal degree, Obstructions to deformation of a complex structure, Orient, 111, 115 Orientability, 111 Orientable bundle, 115 pair, 111 Orientation sheet of the manifold pair (X, F), 111 Oriented pair, fundamental class of, 113 Orienting, co-, 111 Parallel, 99 Periodic cohomology theory, 7 Periodicity, Bott, 7 7r-manifold, 46 Pontrjagin classes, 20 multiplication, 125 numbers, 41 Pontrjagin-Alexander duality, 109 theorem, 124 Primitive left, 167 right, 167 Principal bundles, homology of, 101 Product interior, 133 triple, 105 Projective space, 3 Projector, 94 Pseudogroup, 59 closed of bidifferentiable transformations, 61 infinitesimal, 66 of bidifferentiable, bianalytic or biholomorphic transformations, 64 resolution of the sheaf of vector fields associated with a continuous r (sheaf of T-vectorfields),85 Pseudogroup of differentiable transformations, continuous, 81 invariant Cartan forms for, 85
11 200 INDEX r-frames, 188 bundle of, 188 r-jet, 187 Real analytic, differentiable r-manifold, 64 Rees-Artin lemma, 24 Representation ring completed, 27 completed of a torus, 26 of a compact Lie group, 25 Resolution of the sheaf of vector fields associated with a continuous pseudogroup r (sheaf of T-vectorfields),85 Riemann-Roch theorem, 7, 20 Rigid, 78 Ring Noetherian, 24 of complex vector bundles, 7 of invariants, 27 Saddle surfaces, 182 Sequence, Wang, Sheaf of groups, 65 of vectorfieldsassociated with a continuous pseudogroup T (sheaf of r-vector fields), resolution of, 85 Solvmanifold, 138 Source of (/), 187 Spaces classifying, 7, 28 CUfford-Klein, 156 differentiable family of Clifford-Klein, 159 function, 109 homogeneous, 7, 31 locally affine, 142 projective, 3 structure on topological, 60 Spectral sequence, 7, 16 Spherical image mapping, 181 Spinor representation, 33 Stability of complex structures, 90 Stable germ, 75 locally, 74 Steenrod and Eilenberg axioms, 7 Stiefel-Whitney classes, 20 numbers, 41 Structure almost-product, 94 complex contact, 144 equation of the canonical form, 191 equations of E. Cartan, 186 T-,64 on a topological space, 60 (See Complex) Submanifold, closed relative, 113 Surfaces, 181 saddle, 182 Surgery, 39-42, 44, 46, 54 Surjective, infinitesimally, 75 Suspension, 9 Target of (/), 187 Tietze's Theorem, 119 Todd genus, 21 Topological space, structure on, 60 Topology, a-adic, 24 Torsion, 95, 186 for vector forms, 88 Torsional derivatives, 99 Torus, 26 completed representation ring of, 26 Transformation, coordinate, 118 Transregular, 116 Transverse bundle, 115 Triple product, 105 Trivial locally, 74 Unitary group, 1, 8 Universal Coefficient Theorem, 126 Variability, index of, 72 Variations of a complex structure, 89 Vector bundles, complex, 8 ring of, 7 Vector fields associated with a continuous pseudogroup r (sheaf of r-vector fields), resolution of the sheaf of, 85 r-,67 Vector forms, 87 /-relatedness for, 90 Jacobi identities, 88 torsion for, 88 types of, 88 vertical, 91 Wang sequence, Weyl group, 23 Whitney-Stiefel classes, 20 numbers, 41 BCDEFGHIJ-AMS
12
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